Generation and Asymmetry of Self-Dual Threshold Functions

Abstract

Properties of self-dual threshold functions are discussed because of the importance of self-dual functions in threshold logic. Since any threshold function can be easily converted into or reduced from a positive self-dual threshold function, we will not lose generality in discussion by exploring the properties of positive self-dual threshold functions. First functions generated by additively or subtractively merging two variables of a positive self-dual threshold function are discussed. Expansions of a positive self-dual threshold function with respect to two variables are then shown, and the generation of functions based on them is discussed. The concepts of strongly asymmetrical selfdual threshold functions and its degree are introduced, and the relation of all self-dual threshold functions of fewer variables with strongly asymmetrical ones is shown. The above discussion enables the classification of threshold functions and the relation between threshold functions of n variables and those of more variables to be better seen.